1. Structural Equation Modeling
Evangelia Demerouti, PhD
Utrecht University
Athens,

2. Structure Introduction to SEM Examples Mediation Moderation
Longitudinal data

3. Use of SEM
To test whether theoretical hypothesis about causal relationships fit to empirical data.
It has a confirmatory character (i.e., researcher determines the relationships between the variables)
It tests relationships between observed as well as unobserved, latent variables
It combines regression, factor analysis and analysis of variance.

4. Steps in the utilization of SEM
Development of hypothesis
Construction of path diagram
Specification of model structure
Identification of model structure
Parameter estimation
Evaluation of the results
Modification of the model

5. 1. Hypothesis How are the constructs related to each other
Independent (latent) variables: exogenous ()
Dependent (latent) variables: endogenous ()
Specify the structural model

6. Number steady customers
Time pressure
Performance x1 y1
observed
y1 = a + b x1
,  are hypothetical, abstract constructs that do not exist in reality and which are measured/operationalized through measurement variables/indicators
Job demands
Performance
Time pressure
Cognitive demands
Number sales
Number steady customers 1 1
1 = a + b 1
latent

7. 2. Construction of path diagram
Specify the measurement model (= the empirical indicators of the latent constructs)
Paint the relationships using the connotation of SEM

8. Path diagram – notation SEM
Structural model
Measurement model

9. 3. Specification of model structure
Mathematical specification of the hypotheses using matrices of equities
Rules
errors should be uncorrelated with the latent constructs (otherwise there is another variable which systematically influences the model variable, i.e., incomplete model)
errors should be uncorrelated with each other (otherwise there is a systematic error that influence all independent variables, i.e., method factor)

10. 4. Identification of model structure
Check whether the matrices can be solved, I.e., whether there is enough information from the empirical data to determine the unknown parameters
If n = number of indicators/observed variables
s = n (n + 1) / 2 correlation coefficients or number of equities
If t = number of unknown parameters then t < s (i.e., df > 0)

11. 5. Parameter estimation
The model theoretical correlation matrix  (sigma) has the correlation coefficients which we expect within the data sample if the model is right and the sample is representing the population
The empirical correlation matrix has the (Pearson product-moment) correlation coefficients (rxy) which indicate in how far the relation between two variables x and y resembles a straight line (if one variable increases, the other does also)
Iterative estimations of the correlation coefficients in  tries to minimize the differences between
and R
Theory
Empirical data
The discrepancy between  and R expresses whether theoretical model is acceptable

12. 5. Parameter estimation: Measurement model
Factor analysis explains the correlation among items by assuming an underlying factor
The respective regression coefficient is called lambda () / loading
Egotistic
business
goals d1
Qebgi1 d2
Qebgc1
Latent
variable
ksi1 x1 x2
11
21
Factor loading =
Indicates the extent to which the ratings of items depend on the latent variable

13. 5. Parameter estimation: Structural model
path coefficient = regression weight
= standardized regression coefficient
The path coefficient for
the independent on the
dependent variables is
indicating in how far
 is explained by 
Egotistic
business
goals d1
Qebgi1 d2
Qebgc1 e1
Qepgi1 e2
Qepgc1
Altruistic
Latent
variable
ksi1
eta1
Independent (1)
Dependent (1) x1 x2 y1 y2

14. 6. Evaluation of the results: Total model
The most commonly used model fit statistics is the Chi Square (2) test for association
2 calculates the degree of independence between two variables (i.e. the theoretically expected values vs. the empirical data)
The larger the discrepancy (independence), the sooner 2 becomes significant
Because we are dealing with a measure of misfit, the p-value for 2 should be larger than .05 to decide that the theoretical model fits the data
However, there are many measures of model fit (see next slides), each with their own assumptions and limitations

19. 6. Evaluation of the results: Model parts
Plausibility of parameter estimation
t-value for the estimated parameters showing whether they are different from 0; t > 1.96, p < .05
Chi square difference test

20. 7. Modification of the model
simplify the model (i.e., delete non-significant parameters or parameters with large standard error)
Expand the model (i.e., include new paths using the modification indexes, m > 5.00)

21. Mediation

22. Mediation
We see that X and Y are correlated (this
correlation is referred to here as “c”).
If a third variable mediates the association between X and Y, then after the effects of the mediator are accounted for, “c” will be equal to zero or will be significantly smaller than it was originally.

23. Mediation: Examples Job demands Exhaustion Performance
Causal Thinking Is Implied Here.
The diagram claims that job demands lead to exhaustion and that exhaustion leads to low performance.
For a discussion of causality see:

24. FOUR STEPS To Assess Mediation
Step 1:  Show that the initial variable is correlated with the outcome.
This step establishes that there is an effect that may be mediated.
Job demands
Exhaustion
Performance

25. FOUR STEPS To Assess Mediation
Step 2: Show that the initial variable is correlated with the mediator.
Exhaustion
Job demands
Performance

26. FOUR STEPS To Assess Mediation
Step 3: Show that the mediator affects the outcome variable.
Thus, the initial variable must be controlled in establishing the effect of the mediator on the outcome.
Exhaustion
Job demands
Performance

27. FOUR STEPS To Assess Mediation
At this point we know that all of the variables are associated with each other.
BUT we want to know if the association between the predictor and the outcome is explained by the mediator.
Does the predictor predict the outcome in the same way after the effects of mediator are accounted for?
Exhaustion
Job demands
Performance

28. What to do? Does the direct path carry any water?
To find out add it to the model and
Determine whether the model is better than it was without the direct path
If the path is needed then complete mediation has not occurred
Is the direct path weaker than the zero order?
Run the model with the path coefficient fixed at the value of the zero order r
Compare the results of this analysis to the results of a model in which the path is “free”

33. Chi-square difference test
Model 2 df
2
df
Indirect effects
13.11 5
10.89 1
p < .001
Direct & indirect effects
2.22 4 
Exhaustion partially mediates the relationship between job demands and performance

34. Moderation

35. Moderation Sounds like mediation but is different
Involves Correlations
Involves a “third” variable
Moderation exists when the association between two variables IS NOT THE SAME at all levels of a third variable.
Interaction

36. Example
MODERATION
The association between neuroticism and exhaustion is different for females than it is for males.

40. Chi-square difference test
Model 2 df
2
df
Constrained parameter
5.31 9
0.54 1
n.s.
Free parameter
4.77 8 
Gender does not moderate the relationship between job demands and performance

41. Mediation/Moderation
Moderation does not tell us why one variable is associated to another variable.
Mediation tells us why one variable is associated to another variable.
Moderation tells us when individual differences in one variable are more strongly associated with individual differences in another variable.

42. Longitudinal Data

43. Stability Model Time 1 Time 2 Time 3 Work pressure Work pressure
WHI
WHI
WHI
Exhaustion
Exhaustion
Exhaustion

44. Causality Model Time 1 Time 2 Time 3 Work pressure Work pressure
WHI
WHI
WHI
Exhaustion
Exhaustion
Exhaustion

45. Reversed Causality Model
Time 1
Time 2
Time 3
Work pressure
Work pressure
Work pressure
WHI
WHI
WHI
Exhaustion
Exhaustion
Exhaustion

46. Results: Causality Model
Time 1
Time 2
Time 3
Work pressure
Work pressure
Work pressure
WHI
WHI
WHI
Exhaustion
Exhaustion
Exhaustion

47. Results: Reversed Causality Model
Time 1
Time 2
Time 3
Work pressure
Work pressure
Work pressure
WHI
WHI
WHI
Exhaustion
Exhaustion
Exhaustion

48. Results: Reciprocal Model
Time 1
Time 2
Time 3
Work pressure
Work pressure
Work pressure
WHI
WHI
WHI
Exhaustion
Exhaustion
Exhaustion

49. Goodness-of-Fit Indices for the Alternative WHI Models, N = 3352 df p
GFI
RMSEA
TLI
CFI
M1. Stability Model
773.28
447
.001
.89
.05
.95
M2. Causality Model
685.44
438
.90
.04
.96
.97
M3. Reversed Causality Model
653.62
M4. Reciprocal Model
558.82
429
.91
.03
.98
Null Model
528 --
.19
.20

50. Reciprocal model
Stressor
WHI
Strain

51. Thank you for your attention Email: E.Demerouti@fss.uu.nl
Student version of AMOS to download in